A time series is a sequence of values/observations/data points that are measured typically at successive points in time spaced at uniform time intervals. In other words, the values/observations/data points are observed at evenly spaced time intervals. Time series data may generally be used in pattern recognition, finance, weather forecasting, earthquake prediction, risk management, anomaly detection, tele-traffic prediction, etc. A time series may be stationary or non-stationary. A stationary time series is the time series data comprising a mean and a variance that are constant over time. In one example, yearly average of daily temperature readings in, for example, Mumbai may be considered as the stationary time series. Although temperature readings may vary from one day to another, the yearly averages may stay approximately the same. A non-stationary time series is the time series data comprising a mean and/or a variance that are not constant over time. For example, electrical demand in Mumbai may have a long-term trend as the demand may increase in response to growing population.
Several forecasting methods exists that attempt to predict future values of the time series data based on the past/historical time series data. For example, one of the forecasting methods includes continuing the trend curve smoothly by a straight line. In one example, the forecasting method may include Auto-Regression and Moving Averages (ARIMA). The ARIMA method assumes that each measurement in the time series data is generated by a linear combination of past measurements plus noise. Although the ARIMA is used extensively, the ARIMA method has proved to be inaccurate when used for non-stationary time series data.
In order to predict future values of the time series data, there is an increased thrust to model chaotic and/or turbulent data i.e. time series data that includes extreme values or large deviations. Generally, the time series data arising from risk management, weather predictions, anomaly detection, tele-traffic prediction, etc. includes the extreme values or the large deviations. For predicting future values of the time series data, traditionally most of the extreme values in the past/historical time series data are removed. The extreme values in the past/historical time series data are removed as they are considered to be outliers, and the known methods are applied on remaining time series data. At times, such extreme values may be treated as missing values and may be replaced by most likely values based on the other values. The replacement of the missing values with the most likely values may be referred to as missing value imputation. Although proponents of such replacement/filtering of the values may be justifiable since the approach may capture the general information but there may be loss of information which might have been critical for future predictions. The criticality of the information may be noticed since the extreme values in essence captures the characteristics of a system that generates the time series data. For example, the extreme values in tele-traffic may occur and may comprise values that may be important and may need to be modelled for capacity planning.
On the other hand, traditional methods generally employ Extreme Value (EV) Distributions or mixture of distributions to capture the extreme values. The future values may be estimated by employing Expectation-maximization (EM), Maximum-Likelihood Estimation (MLE) or Bayesian methods. While employing the above methods, the extreme values are not removed from the time series data; instead the extreme values are utilized to build models without loss of any information. Further, in order to generate future forecasts either mixed or multiple data distributions are employed instead of using a single data distribution. In addition, for parameter estimation MLE, Bayesian or EM algorithm is applied. However, the methods discussed neither addresses the expected change or the rate of change in time series data. It is important to understand the rate of change of change in time series data. For example, in case of financial market data, it may be important to predict the expected future change in the market data. Further, most of the time series data involving the extreme values are generally non-stationary. Therefore, when the time series data is integrated, distributional assumptions related to Data Generating Process (DGP), for example mixture of Gaussians requires re-parameterization. If non-Gaussian distributions are employed then re-parameterization may get substantially complicated, for example text arrival in streaming data that is modelled as Dirichlet process. Further, existing methods do not provide both point and probability estimates of the forecasted value.